Explicit class field theory for global function fields

Let F be a global function field and let Fab be its maximal abelian extension. Following an approach of D. Hayes, we shall construct a continuous homomorphism ρ:Gal(Fab/F)→CF, where CF is the idele class group of F. Using class field theory, we shall show that our ρ is an isomorphism of topological groups whose inverse is the Artin map of F. As a consequence of the construction of ρ, we obtain an explicit description of Fab. Fix a place ∞ of F, and let A be the subring of F consisting of those elements which are regular away from ∞. We construct ρ by combining the Galois action on the torsion points of a suitable Drinfeld A-module with an associated ∞-adic representation studied by J.-K. Yu.